In this section, we present a non-asymptotic analysis for an extensive list of key quantities in GP regression, encompassing the posterior mean, the posterior variance, and their derivative counterparts, which is based on an operator-theoretic approach [1]}, [2]} and the equivalent kernel technique. Our non-asymptotic analysis applies to general Mercer kernels with uniformly bounded eigenfunctions (Condition (A1)). We substantially extend the analysis in our earlier work [3]} by deriving a range of either new or sharper error bounds, which are critical to conclude minimax optimality and the equivalence theory for the regression function and differential operators. As a byproduct, our analysis provides a series of convergence rates for point estimators of the regression function and its derivatives, and characterizes uncertainty in GP regression by bounding its posterior variances. This non-asymptotic framework may also be of independent interest.
